Mathematics Standard • Year 11 • Module 1 • Lesson 8

Formula and Equation Synthesis

Build fluency choosing the right first move, substitute, solve, rearrange or build, for mixed practical algebra questions.

Build · Skill Drill

1. Quick recall, name the strategy

For each prompt, write S (substitute), E (solve equation), R (rearrange) or B (build). 1 mark each

Q1.1 "Use C = 12 + 4r to find C when r = 7." Strategy: ____________

Q1.2 "12 + 4r = 40. Find r." Strategy: ____________

Q1.3 "d = st. Find s when d = 135 and t = 3." Strategy: ____________

Q1.4 "A table shows outputs going up by 5 each step. Write a formula." Strategy: ____________

Stuck? Revisit lesson § Choose the First Move, match the question type to the best first move.

2. Worked example, choose then solve

Follow each line of working. Note how the first decision (which strategy?) drives every step that follows.

Problem. A printer charges $25 setup plus $2 per page. A job costs $81. How many pages were printed?

Step 1, Choose strategy.

Total known, pages unknown ⇒ Solve an equation.

Reason: we have the output ($81) and want the input (number of pages).

Step 2, Define the variable.

Let p = number of pages.

Step 3, Write and solve the equation.

25 + 2p = 81 ⇒ 2p = 56 ⇒ p = 28

Reason: subtract 25 from both sides, then divide both sides by 2.

Step 4, Reasonableness check.

Check: 25 + 2(28) = 25 + 56 = 81 ✓

Conclusion. 28 pages were printed.

3. Faded example, rearrange first

Use d = st to find the average speed when d = 135 km and t = 3 h. Fill in each blank. 4 marks

Step 1, Choose strategy:

Speed is not the subject of d = st, so we ____________ before substituting.

Step 2, Rearrange:

d = st ⇒ s = ____ ÷ ____

Step 3, Substitute:

s = ____ ÷ ____ = ____________

Conclusion sentence. The average speed is ____________ km/h.

Stuck? Revisit lesson § Worked Example 2, Rearrange before substituting. To make s the subject, divide both sides by t.

4. Graduated practice, pick the strategy, then solve

For every question, name the strategy first (S / E / R / B), then show the working.

Foundation, clean numbers (4 questions)

Q	Problem	Strategy & answer
4.1 1	Use C = 12 + 4r to find C when r = 7.
4.2 1	Solve 3x + 5 = 23 for x.
4.3 1	Use A = lw. Find l when A = 60 and w = 5.
4.4 1	A table: input 0,1,2,3 → output 4,9,14,19. Write a formula.

Standard, typical HSC difficulty (6 questions)

Show choice of strategy, working, and a units sentence in the conclusion.

4.5 A gym charges $20 plus $15 per class. Find the number of classes if the total is $110. 2 marks

4.6 Use A = s² to find A when s = 9.5 m. 2 marks

4.7 Rearrange A = bh to find h. Then find h when A = 72 cm² and b = 9 cm. 2 marks

4.8 Write a formula for outputs 8, 13, 18, 23 for inputs 0, 1, 2, 3. Test it using input 3. 3 marks

4.9 A hire company charges $35 + $18 per hour. The total cost is $143. Find the hire time. 2 marks

4.10 Use d = st to find time when d = 210 km and s = 70 km/h. Rearrange first. 2 marks

Extension, strategy + reasonableness (2 questions)

4.11 A stopping-distance model is D = 0.01v² + 0.3v. A student claims that at 60 km/h the stopping distance is 540 m. Calculate D at v = 60 and explain what is wrong with the student's answer. 3 marks

4.12 A table shows distance vs time: at t = 0 h, d = 12 km; at t = 1 h, d = 92 km; at t = 2 h, d = 172 km. Write a formula linking d and t, then use it to predict d at t = 3.5 h. 4 marks

Stuck on 4.11? Substitute v = 60 carefully and compare. The student likely multiplied 0.01 by 60 then added v² in the wrong order.

5. Self-check the easy 3

Tick the first three once you've checked your method works.

For 4.1 I substituted (S) to get C = 12 + 4(7) = 40.

For 4.2 I solved (E): 3x = 18, x = 6.

For 4.3 I rearranged (R) A = lw → l = A/w = 60/5 = 12.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers, Do not peek before attempting

Q1.1–Q1.4, Strategy names

Q1.1: S (Substitute, output unknown). Q1.2: E (Solve equation, input unknown). Q1.3: R (Rearrange, wrong subject). Q1.4: B (Build a formula from the table pattern).

Q3, Faded speed example

Step 1: We rearrange before substituting.
Step 2: s = d ÷ t.
Step 3: s = 135 ÷ 3 = 45.
Conclusion: The average speed is 45 km/h.

Q4.1, C = 12 + 4r at r = 7 (S)

C = 12 + 4(7) = 12 + 28 = 40.

Q4.2, Solve 3x + 5 = 23 (E)

3x = 18, x = 6.

Q4.3, A = lw, find l (R)

l = A/w = 60/5 = 12.

Q4.4, Build formula (B)

Start = 4, repeated increase = +5 per input step. y = 4 + 5x. Check x = 3: 4 + 5(3) = 19 ✓.

Q4.5, Gym (E)

Let c = classes. 20 + 15c = 110 → 15c = 90 → c = 6 classes.

Q4.6, A = s² at s = 9.5 (S)

A = (9.5)² = 90.25 m².

Q4.7, A = bh, find h (R)

h = A/b = 72/9 = 8 cm.

Q4.8, Build formula (B)

Start = 8 (at input 0), repeated increase = +5. y = 8 + 5x. Test x = 3: 8 + 5(3) = 23 ✓.

Q4.9, Hire company (E)

Let h = hours. 35 + 18h = 143 → 18h = 108 → h = 6 hours.

Q4.10, d = st, find t (R)

t = d/s = 210/70 = 3 h.

Q4.11, Reasonableness check on D = 0.01v² + 0.3v at v = 60

D = 0.01(60)² + 0.3(60) = 0.01(3600) + 18 = 36 + 18 = 54 m. The student's 540 m is ten times too large likely 0.1 used instead of 0.01, or a misplaced decimal point.

Q4.12, Build and use a formula (B + S)

Start (t = 0): d = 12. Repeated change: +80 km per hour. Formula: d = 12 + 80t. Predict t = 3.5: d = 12 + 80(3.5) = 12 + 280 = 292 km.